Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes

Holzegel, Gustav; Warnick, Claude

doi:10.1016/j.jfa.2013.10.019

Cited by 35 publications

(104 citation statements)

References 25 publications

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“…This superpolynomial decay in the exterior is sufficient so as to follow the method of [26] with vector fields of the form (1.2) to show boundedness and continuity at the Cauchy horizon, up to the additional difficulty caused by the fact that we allow a possibly negative Klein-Gordon mass parameter. The violation of the dominant energy condition due to the presence of a negative mass term can be overcome with twisted derivatives [63,42], which provide a useful framework to replace Hardy inequalities for the lower order terms in this context.…”

Section: Introductionmentioning

confidence: 99%

Uniform Boundedness and Continuity at the Cauchy Horizon for Linear Waves on Reissner–Nordström–AdS Black Holes

Kehle

2019

Commun. Math. Phys.

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Motivated by the Strong Cosmic Censorship Conjecture for asymptotically AdS spacetimes, we initiate the study of massive scalar waves satisfying g ψ − µψ = 0 on the interior of Anti-de Sitter (AdS) black holes. We prescribe initial data on a spacelike hypersurface of a Reissner-Nordström-AdS black hole and impose Dirichlet (reflecting) boundary conditions at infinity. It was known previously that such waves only decay at a sharp logarithmic rate (in contrast to a polynomial rate as in the asymptotically flat regime) in the black hole exterior. In view of this slow decay, the question of uniform boundedness in the black hole interior and continuity at the Cauchy horizon has remained up to now open. We answer this question in the affirmative.Conjecture 1 (C 0 formulation of strong cosmic censorship). For generic compact or asymptotically flat (asymptotically Anti-de Sitter) vacuum initial data, the maximal Cauchy development of (EE) is inextendible as a Lorentzian manifold with C 0 (continuous) metric.Surprisingly, the C 0 formulation (Conjecture 1) was recently proved to be false for both cases Λ = 0 and Λ > 0 (see discussion later, [17]). However, the following weaker, yet well-motivated, formulation introduced by Christodoulou in [9] is still expected to hold true (at least) in the asymptotically flat case (Λ = 0).Conjecture 2 (Christodoulou's re-formulation of strong cosmic censorship). For generic asymptotically flat vacuum initial data, the maximal Cauchy development of (EE) is inextendible as a Lorentzian manifold with C 0 (continuous) metric and locally square integrable Christoffel symbols.1 More precisely, this holds true for subextremal and non-trivially rotating(charged) Kerr(Reissner-Nordström) black holes which we will assume for the rest of the paper, unless explicitly stated otherwise.In order to gain insight about SCC, the most naive approach (often referred to as "poor man's linearization") is to study solutions of (1.1) with µ = 0 on a fixed explicit black hole spacetime (e.g. Kerr or Reissner-Nordström). This can be considered as the most naive toy model for (EE) with initial data close to Kerr or Reissner-Nordström data, for which many features of (EE) including the non-linear terms and the tensorial structure are neglected; see the pioneering works for asymptotically flat (Λ = 0) black holes [60,48,49,8]. Under the identification ψ ∼ g and ∂ψ ∼ Γ, where ψ is a solution to (1.1), Conjecture 1 corresponds to a failure of ψ to be continuous (C 0 ) at the Cauchy horizon. Similarly, Conjecture 2 corresponds to a failure of ψ to lie in H 1 loc at the Cauchy horizon. The state of the art for Λ = 0 and Λ > 0. The definitive disproof [17] of Conjecture 1 was preceded by corresponding results on the level of (1.1).Linear level for Λ = 0. In the asymptotically flat case (Λ = 0) it was shown in [26,27] (see also [34]) that solutions of (1.1) with µ = 0 arising from data on a spacelike hypersurface remain continuous and uniformly bounded (no C 0 blow-up) at the Cauchy horizon of general subextremal Kerr or...

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Section: Introductionmentioning

confidence: 99%

Uniform Boundedness and Continuity at the Cauchy Horizon for Linear Waves on Reissner–Nordström–AdS Black Holes

Kehle

2019

Commun. Math. Phys.

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“…In [16], Holzegel and Smulevici proved that for solutions to the Klein-Gordon equation on Kerr-AdS, a non-degenerate energy decays slowly in time (as an inverse power of the logarithm). In [17], Holzegel and Warnick studied the linear stability (in the sense of uniform boundedness of solutions of the Klein-Gordon equation) of stationary AdS black holes in general, and of the spherically symmetric AdS Schwarzschild black hole in particular. The full non-linear (orbital and asymptotic) stability of the spherical AdS Schwarzschild black hole was proved in [18] within the class of spherical symmetry.…”

Section: Background and Motivationmentioning

confidence: 99%

“…In the final section of the paper, we follow the methods of [17] and particularly [19] to prove a polynomial decay rate for a non-degenerate energy of solutions of the Klein-Gordon equation when M > 0 and 1/2 ≥ κ > 0. Note that this includes the conformally coupled case κ = 1/2.…”

Section: Energy Decaymentioning

confidence: 99%

“…in the particular case that k = −1, on the exterior of the black hole, and where α satisfies the Breitenlohner-Friedman bound 0 < α < 9/4 [8]. The cases when k = 1, 0 have been studied already in [17] and [19] respectively.…”

Section: The Space-timementioning

confidence: 99%

“…It is clearly a symmetric tensor of type (0, 2) but unlike the usual energy momentum tensor it is not in general divergence free. The following basic properties of the twisted energy momentum tensor were proved in [17].…”

Section: The Twisted Energy-momentum Tensormentioning

confidence: 99%

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The Klein–Gordon equation on the hyperboloidal anti-de Sitter Schwarzschild black hole

Fitz-Gibbon¹

2019

Class. Quantum Grav.

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In this paper we establish energy decay for solutions to the Klein-Gordon equation on the positive mass hyperboloidal anti-de Sitter Schwarzschild black hole, subject to Dirichlet, Neumann and Robin boundary conditions at infinity, for a range of the (negative) mass squared parameter. To do so we use vector field methods with a renormalised energy to avoid divergences that would otherwise appear in the energy integrals. For another region of the parameter space, we use the existence of negative energy solutions to demonstrate linear instability.

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Decay Properties of Klein‐Gordon Fields on Kerr‐AdS Spacetimes

Holzegel

Smulevici

2013

Comm Pure Appl Math

191

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Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes

Cited by 35 publications

References 25 publications

Uniform Boundedness and Continuity at the Cauchy Horizon for Linear Waves on Reissner–Nordström–AdS Black Holes

Uniform Boundedness and Continuity at the Cauchy Horizon for Linear Waves on Reissner–Nordström–AdS Black Holes

The Klein–Gordon equation on the hyperboloidal anti-de Sitter Schwarzschild black hole

Decay Properties of Klein‐Gordon Fields on Kerr‐AdS Spacetimes

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